Understanding the Problem: Interchanging Numbers in Equations

The question asks us to examine four mathematical equations. For each equation, we need to swap the positions of the numbers 3 and 4. After performing this interchange, we must evaluate the modified equation using the correct order of operations (like BODMAS or PEMDAS) and check if the resulting value matches the original result given in the equation. The task is to identify which of the given equations becomes incorrect after this interchange of 3 and 4.

To solve this, we will go through each option one by one, perform the interchange, and then evaluate the equation.

Order of Operations (BODMAS/PEMDAS)

Remember the order of operations to correctly evaluate mathematical expressions:

• B/P: Brackets/Parentheses first
• O/E: Orders/Exponents (powers, square roots) next
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

Analyzing Each Equation by Interchanging 3 and 4

Option 1 Analysis: Check Equation Correctness

The original equation is: \(3 + 5 \times 4 - 6 \div 2 = 16\)

Now, we interchange 3 and 4. The new equation becomes:

\(4 + 5 \times 3 - 6 \div 2\)

Let's evaluate this using BODMAS:

• First, perform Multiplication and Division from left to right:
  • \(5 \times 3 = 15\)
  • \(6 \div 2 = 3\)
• The equation is now: \(4 + 15 - 3\)
• Next, perform Addition and Subtraction from left to right:
  • \(4 + 15 = 19\)
  • \(19 - 3 = 16\)

The result after interchanging 3 and 4 is 16.

Comparing with the original result (16), we see that \(16 = 16\).

Conclusion for Option 1: The equation remains correct after interchanging 3 and 4.

Option 2 Analysis: Identifying the Incorrect Equation

The original equation is: \(4 \times 6 - 3 + 9 \div 1 = 25\)

Now, we interchange 3 and 4. The new equation becomes:

\(3 \times 6 - 4 + 9 \div 1\)

Let's evaluate this using BODMAS:

• First, perform Multiplication and Division from left to right:
  • \(3 \times 6 = 18\)
  • \(9 \div 1 = 9\)
• The equation is now: \(18 - 4 + 9\)
• Next, perform Addition and Subtraction from left to right:
  • \(18 - 4 = 14\)
  • \(14 + 9 = 23\)

The result after interchanging 3 and 4 is 23.

Comparing with the original result (25), we see that \(23 \neq 25\).

Conclusion for Option 2: The equation becomes incorrect after interchanging 3 and 4.

Option 3 Analysis: Check Equation Correctness

The original equation is: \(3 \times 7 - 8 \div 2 + 4 = 27\)

Now, we interchange 3 and 4. The new equation becomes:

\(4 \times 7 - 8 \div 2 + 3\)

Let's evaluate this using BODMAS:

• First, perform Multiplication and Division from left to right:
  • \(4 \times 7 = 28\)
  • \(8 \div 2 = 4\)
• The equation is now: \(28 - 4 + 3\)
• Next, perform Addition and Subtraction from left to right:
  • \(28 - 4 = 24\)
  • \(24 + 3 = 27\)

The result after interchanging 3 and 4 is 27.

Comparing with the original result (27), we see that \(27 = 27\).

Conclusion for Option 3: The equation remains correct after interchanging 3 and 4.

Option 4 Analysis: Check Equation Correctness

The original equation is: \(9 \times 3 \div 6 + 4 - 8 = 1\)

Now, we interchange 3 and 4. The new equation becomes:

\(9 \times 4 \div 6 + 3 - 8\)

Let's evaluate this using BODMAS:

• First, perform Multiplication and Division from left to right:
  • \(9 \times 4 = 36\)
  • \(36 \div 6 = 6\)
• The equation is now: \(6 + 3 - 8\)
• Next, perform Addition and Subtraction from left to right:
  • \(6 + 3 = 9\)
  • \(9 - 8 = 1\)

The result after interchanging 3 and 4 is 1.

Comparing with the original result (1), we see that \(1 = 1\).

Conclusion for Option 4: The equation remains correct after interchanging 3 and 4.

Summary of Results after Interchanging 3 and 4

Based on our analysis, the equation in Option 2 is the only one that is not correct after interchanging the numbers 3 and 4.

Conclusion

By carefully interchanging 3 and 4 and applying the order of operations (BODMAS/PEMDAS) to each equation, we found that only the equation from Option 2 resulted in a value different from the original equation's result. Therefore, the equation in Option 2 will be not correct after interchanging the given two numbers.

Revision Table: Understanding Interchanging Numbers

Additional Information: Solving Equations with Number Swaps

Problems involving interchanging numbers are common in tests of numerical ability and logical reasoning. They require careful substitution and precise application of mathematical rules like the order of operations. When tackling such problems:

• Read the instruction carefully: Which numbers are being interchanged? Where?
• Rewrite the expression or equation after the interchange.
• Strictly follow the order of operations (BODMAS/PEMDAS) during evaluation.
• Perform calculations step-by-step to minimize errors.
• Compare the final result with the target value (the original result in this case) to determine correctness.
• These problems test attention to detail and fundamental arithmetic skills.